**Solution:**

The expression |x−1|−1 will equal 0 if |x−1|=1. This is true for x=2 and for x=0. For example, substituting x=2 into the expression |x−1|−1 and simplifying the result yields |2−1|−1=|1|−1=1−1=0.Therefore, there is a value of x for which |x−1|−1 is equal to 0.

Choice B is incorrect. By definition, the absolute value of any expression is a non-negative number. Substituting any value for x into the expression |x + 1| will yield a nonnegative number as the result. Because the sum ofa non-negative number and a positive number is positive, |x + 1| + 1 will be a positive number for any value of x. Therefore,|x+1|+1≠0 for any value of x. Choice C is incorrect. By definition, the absolute value of any expression is a non-negative number. Substituting any value for x into the expression |1−x| will yield a non-negative number as the result. Because the sum of a non-negative number and a positive number is positive, |1−x|+1 will be a positive number for any value of x. Therefore,|1−x|+1≠0 for any value of x. Choice D is incorrect. By definition, the absolute value of any expression is a non-negative number. Substituting any value for x into the expression |x−1| will yield a non-negative number as the result. Because the sum o1f a non-negative number and a positive number is positive,|x−1|+ will be a positive number for any value of x. Therefore, |x−1|+1≠0 for any value of x.

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